As a first attempt, we fit a simple Poisson regression:
\[
ln\lambda_i = \alpha + \beta\cdot elapsed\_time_i \\
y_i \sim \mathcal{Poisson}(\lambda_i)
\]
with \(i = 1,\dots,83\), and \(y_i\) represents the number of cases.
##
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## Inference for Stan model: poisson_regression.
## 4 chains, each with iter=2000; warmup=1000; thin=1;
## post-warmup draws per chain=1000, total post-warmup draws=4000.
##
## mean se_mean sd 2.5% 25% 50% 75% 97.5% n_eff Rhat
## alpha 0.18 0 0.13 -0.07 0.10 0.18 0.26 0.43 897 1.01
## beta 0.12 0 0.01 0.11 0.11 0.12 0.12 0.13 893 1.00
##
## Samples were drawn using NUTS(diag_e) at Thu Jul 2 22:23:27 2020.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at
## convergence, Rhat=1).
Looking at Rhat we can see that we have reached the convergence.
theme_set(bayesplot::theme_default())
mcmc_scatter(as.matrix(fit.model.Poisson, pars=c("alpha", "beta") ), alpha=0.2)